1. Advanced Normalization
Chapter 15
Advanced Normalization
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2. Chapter 15 - Objectives
How inference rules can identify a set of all functional dependencies for a relation.
How Inference rules called Armstrong’s axioms can identify a minimal set of useful functional dependencies from the set of all functional dependencies for a relation.
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3. Chapter 15 - Objectives
Normal forms that go beyond Third Normal Form (3NF), which includes Boyce-Codd Normal Form (BCNF), Fourth Normal Form (4NF), and Fifth Normal Form (5NF).
How to identify Boyce–Codd Normal Form (BCNF).
How to represent attributes shown on a report as BCNF relations using normalization.
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4. Chapter 15 - Objectives
Concept of multi-valued dependencies and Fourth Normal Form (4NF).
The problems associated with relations that break the rules of 4NF.
How to create 4NF relations from a relation, which breaks the rules of to 4NF.
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5. Chapter 15 - Objectives
Concept of join dependency and Fifth Normal Form (5NF).
The problems associated with relations that break the rules of 5NF.
How to create 5NF relations from a relation, which breaks the rules of 5NF.
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6. More on Functional Dependencies
The complete set of functional dependencies for a given relation can be very large.
Important to find an approach that can reduce the set to a manageable size.
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7. Inference Rules for Functional Dependencies
Need to identify a set of functional dependencies (represented as X) for a relation that is smaller than the complete set of functional dependencies (represented as Y) for that relation and has the property that every functional dependency in Y is implied by the functional dependencies in X.
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8. Inference Rules for Functional Dependencies
The set of all functional dependencies that are implied by a given set of functional dependencies X is called the closure of X, written X+ .
A set of inference rules, called Armstrong’s axioms, specifies how new functional dependencies can be inferred from given ones.
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9. Inference Rules for Functional Dependencies
Let A, B, and C be subsets of the attributes of the relation R. Armstrong’s axioms are as follows:
 (1) Reflexivity
If B is a subset of A, then A → B
(2) Augmentation
If A → B, then A,C → B,C
(3) Transitivity
If A → B and B → C, then A → C
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10. Inference Rules for Functional Dependencies
Further rules can be derived from the first three rules that simplify the practical task of computing X+. Let D be another subset of the attributes of relation R, then:
(4) Self-determination
A → A
(5) Decomposition
If A → B,C, then A → B and A → C
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11. Inference Rules for Functional Dependencies
(6) Union
If A → B and A → C, then A → B,C
(7) Composition
If A → B and C → D then A,C → B,D
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12. Minimal Sets of Functional Dependencies
A set of functional dependencies Y is covered by a set of functional dependencies X, if every functional dependency in Y is also in X+; that is, every dependency in Y can be inferred from X.
A set of functional dependencies X is minimal if it satisfies the following conditions:
Every dependency in X has a single attribute on its right-hand side.
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13. Minimal Sets of Functional Dependencies
We cannot replace any dependency A → B in X with dependency C → B, where C is a proper subset of A, and still have a set of dependencies that is equivalent to X.
We cannot remove any dependency from X and still have a set of dependencies that is equivalent to X.
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14. Boyce–Codd Normal Form (BCNF)
Based on functional dependencies that take into account all candidate keys in a relation, however BCNF also has additional constraints compared with the general definition of 3NF.
Boyce–Codd normal form (BCNF)
A relation is in BCNF if and only if every determinant is a candidate key.
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15. Boyce–Codd Normal Form (BCNF)
Difference between 3NF and BCNF is that for a functional dependency A  B, 3NF allows this dependency in a relation if B is a primary-key attribute and A is not a candidate key. Whereas, BCNF insists that for this dependency to remain in a relation, A must be a candidate key.
Every relation in BCNF is also in 3NF. However, a relation in 3NF is not necessarily in BCNF.
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16. Boyce–Codd Normal Form (BCNF)
Violation of BCNF is quite rare.
The potential to violate BCNF may occur in a relation that:
contains two (or more) composite candidate keys;
the candidate keys overlap, that is have at least one attribute in common.
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17. Review of Normalization (UNF to BCNF)
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18. Review of Normalization (UNF to BCNF)
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19. Review of Normalization (UNF to BCNF)
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20. Review of Normalization (UNF to BCNF)
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21. Fourth Normal Form (4NF)
Although BCNF removes anomalies due to functional dependencies, another type of dependency called a multi-valued dependency (MVD) can also cause data redundancy.
Possible existence of multi-valued dependencies in a relation is due to 1NF and can result in data redundancy.
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22. Fourth Normal Form (4NF)
Multi-valued Dependency (MVD)
Dependency between attributes (for example, A, B, and C) in a relation, such that for each value of A there is a set of values for B and a set of values for C. However, the set of values for B and C are independent of each other.
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23. Fourth Normal Form (4NF)
MVD between attributes A, B, and C in a relation using the following notation:
A −>> B
A −>> C
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24. Fourth Normal Form (4NF)
A multi-valued dependency can be further defined as being trivial or nontrivial.
A MVD A −>> B in relation R is defined as being trivial if (a) B is a subset of A or (b) A  B = R.
A MVD is defined as being nontrivial if neither (a) nor (b) are satisfied.
A trivial MVD does not specify a constraint on a relation, while a nontrivial MVD does specify a constraint.
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25. Fourth Normal Form (4NF)
Defined as a relation that is in Boyce-Codd Normal Form and contains no nontrivial multi-valued dependencies.
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26. 4NF - Example
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27. Fifth Normal Form (5NF)
A relation decompose into two relations must have the lossless-join property, which ensures that no spurious tuples are generated when relations are reunited through a natural join operation.
However, there are requirements to decompose a relation into more than two relations. Although rare, these cases are managed by join dependency and fifth normal form (5NF).
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28. Fifth Normal Form (5NF)
Defined as a relation that has no join dependency.
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29. 5NF - Example
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30. 5NF - Example
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